Limit theorems for quadratic forms of Lévy-driven continuous-time linear processes

Shuyang Bai, Mamikon S. Ginovyan and Murad S. Taqqu

Stochastic Processes and their Applications, 2016, vol. 126, issue 4, 1036-1065

Abstract: We study the asymptotic behavior of a suitable normalized stochastic process {QT(t),t∈[0,1]}. This stochastic process is generated by a Toeplitz type quadratic functional of a Lévy-driven continuous-time linear process. We show that under some Lp-type conditions imposed on the covariance function of the model and the kernel of the quadratic functional, the process QT(t) obeys a central limit theorem, that is, the finite-dimensional distributions of the standard T normalized process QT(t) tend to those of a normalized standard Brownian motion. In contrast, when the covariance function of the model and the kernel of the quadratic functional have a slow power decay, then we have a non-central limit theorem for QT(t), that is, the finite-dimensional distributions of the process QT(t), normalized by Tγ for some γ>1/2, tend to those of a non-Gaussian non-stationary-increment self-similar process which can be represented by a double stochastic Wiener–Itô integral on R2.

Keywords: Toeplitz type quadratic functional; Lévy process; Brownian motion; Central and non-central limit theorems; Long memory; Teugels martingale; Wiener–Itô integral (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2015.10.010

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